TY - JOUR

T1 - Stratified periodic water waves with singular density gradient

AU - Escher, Joachim

AU - Knopf, Patrik

AU - Lienstromberg, Christina

AU - Matioc, Bogdan-Vasile

N1 - Funding Information: Open Access funding provided by Projekt DEAL. Patrik Knopf and Bogdan-Vasile Matioc were partially supported by the RTG 2339 “Interfaces, Complex Structures, and Singular Limits” of the German Science Foundation (DFG). Christina Lienstromberg has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the collaborative research centre ‘The mathematics of emerging effects’ (CRC 1060, Projekt-ID 211504053 ) and the Hausdorff Center for Mathematics (GZ 2047/1, Projekt-ID 390685813). The support is gratefully acknowledged.

PY - 2020/10

Y1 - 2020/10

N2 - We consider Euler’s equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each crest and trough. Our analysis uses, to a large extent, the availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis.

AB - We consider Euler’s equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each crest and trough. Our analysis uses, to a large extent, the availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis.

KW - math.AP

KW - Stratified fluid

KW - Singular density gradient

KW - Traveling waves

KW - Euler equations

UR - http://www.scopus.com/inward/record.url?scp=85079445593&partnerID=8YFLogxK

U2 - 10.1007/s10231-020-00950-1

DO - 10.1007/s10231-020-00950-1

M3 - Article

VL - 199

SP - 1923

EP - 1959

JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

SN - 0373-3114

IS - 5

ER -